An unfolding of the Takens-Bogdanov singularity
Authors:
P. Hirschberg and E. Knobloch
Journal:
Quart. Appl. Math. 49 (1991), 281-287
MSC:
Primary 58F14; Secondary 34C23
DOI:
https://doi.org/10.1090/qam/1106393
MathSciNet review:
MR1106393
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Abstract: A complete analysis for $\left | \mu \right |, \left | \nu \right | \ll 1$ of the equation \[ \dot x = y, \dot y = \mu x + \nu y + M{x^2} + \Gamma xy\] describing a particular unfolding of the Takens-Bogdanov singularity is presented.
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- William J. Merryfield, Juri Toomre, and Douglas Gough, Nonlinear behavior of solar gravity modes driven by $^3{\rm He}$ in the core. I. Bifurcation analysis, Astrophys. J. 353 (1990), no. 2, 678–697. MR 1049653, DOI https://doi.org/10.1086/168658
- Floris Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 47–100. MR 339292
R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Functional Anal. Appl. (2) 9, 144–145 (1975)
J. Carr, Applications of Center Manifold Theory, Springer-Verlag, New York-Heidelberg-Berlin, 1981
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York-Heidelberg-Berlin, 1986
B. Keyfitz, Admissibility conditions for shocks in conservation laws that change type, SIAM J. Math. Anal., in press (1991)
B. Keyfitz, Shocks near the sonic line: A comparison between steady and unsteady models for change of type, Nonlinear Evolution Equations that Change Type, B. L. Keyfitz and M. Shearer, eds., Springer-Verlag, New York-Heidelberg-Berlin, 1990
E. Knobloch and M. R. E. Proctor, Nonlinear periodic convection in double-diffusive systems, J. Fluid Mech. 108, 291–316 (1981)
N. Kopell and L. N. Howard, Bifurcations and trajectories connecting critical points, Adv. Math. (3) 18, 306–358 (1975)
W. Merryfield, J. Toomre, and D. Gough, Nonlinear behavior of solar gravity modes driven by $^{3}$He in the core I. Bifurcation analysis, Astrophys. J. 353, 678–697 (1990)
F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math. 43, 47–100 (1974)
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Article copyright:
© Copyright 1991
American Mathematical Society
